Aircraft: EMB ERJ 170/175 ER/LR
Phase of flight: Climb
Anomaly: Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT

Clearance was depart Runway XX SID; but intercept airway; not to track to ZZZ VOR; maintain 8;000 ft. I briefed my Captain (his leg) on the clearance and he said it was normal during clear weather to depart on the SID; only to be given a direct to clearance in the turn. Off the crosswind leg; we selected direct ZZZZZ; as it offered the most logical track with optimal terrain clearance. It was hot; but we were climbing well at 73K Lbs. At around 4;100 ft. we began accelerating to 250 kts. and the rate of climb slowed to 500 FPM. There was no green; yellow; or red terrain in the map. It was all black; meaning not a threat. We recieved a radio call from ATC; giving as an altitude alert and querying us of possibly leveling at 4;000 ft. (we were cleared to 15;000 ft.; which was in the alerter). At that point we were in a slower climb as we were accelerating to 250. He directed us to expedite our climb for terrain and further turned us 30 degrees right; although we were well clear of everything. Captain selected man speed 210 kts. and increased our rate of climb. 10-15 seconds later; center reported clear of all terrain and we resumed on coarse. A few minutes later; center advised us of a possible deviation and passed a number to call. We were VMC and clear for the entire evolution; which lasted about 30 seconds.In hindsight; I will either fly the full SID if there's any question (at a minimum; clarify with clearance delivery); or possibly request a heading departure if that's what it sounds like I'm doing anyway. I think I might have focused too much on the 'intercept the V386' verbiage when they were possibly expecting us to fly a climbing course reversal procedure over a VOR; as depicted. Sometimes what's most logical; economical; or 'how it's historically flown' is actually a recipe for a violation. I don't think this event would have been an issue; except that the ATC software triggered an alarm; based on the decrease in climb rate (as the aircraft decreased pitched momentarily - AP Engaged - in an effort to attain 250 kts.) with the associated terrain in the distance. Slowing increasing the airspeed in manual fashion; while keeping the rate of climb consistently between 2;800-4;000 fpm would have kept us trending in a conflict-free direction. It was only my second time into ZZZ and it bit me. It's a special airport for a reason and this a great example of why every crew has to bring their A game; every time. We were initially cleared the SID via ZZZ; then given the SID; but not to ZZZ; but direct to intercept airway. We selected ZZZZZ; as it's the lowest terrain. It was hot but we were climbing well. At around 4;100 ft. we began accelerating to 250 kts. and the rate of climb slowed to 500 FPM. There was no green; yellow; or red terrain in the map. It was all black; meaning not a threat. We got a radio call from ATC; giving as an altitude alert and querying us of possibly leveling at 4;000 ft. (we were cleared to 15;000 ft.; which was in the alerter) at that point and were in the slow climb. He directed us to expedite our climb for terrain and further turn 30 degrees right for terrain; although we were well clear of it. Later center gave me a number to call. We were also VMC AND CLEAR.If ATC is not comfortable with us maintaining our own terrain clearance they should keep us on the SID. E175 flight crew reported 'low altitude alert' from ATC during a mountainous terrain departure.